Ordinary construction and cleanout debris is fine in a rented container. What cannot go in comes from Missouri rules and from the city's own definitions.
Belton keeps a list of banned and exempt material that stays out of the regular waste stream. It covers hazardous waste, contaminated waste, industrial waste, sludge, biological and medical waste, lead-acid batteries, used oil, used tires, automobile parts, boats and boat trailers, and internal combustion engines. Those need their own handling, and your account manager can point you to it.
Liquids are out as well. The city requires waste to be drained of all liquids before it goes in a container, and loads have to travel covered so nothing escapes on the road.
Construction and demolition debris sits in an interesting spot here. The city's own curbside and bulky collection will not take it at all, which is exactly why a rented container is the usual answer for a remodel. It still has to go to a facility the state has approved to receive it, which is handled on our end.
If you are unsure about an item, ask your account manager before it goes in the container.